Trajectory determination for four-wheel steering

Abstract

Four-wheel steering of a vehicle, e.g., in which leading wheels and trailing wheels are steered independently of each other, can provide improved maneuverability and stability. A first vehicle model may be used to determine trajectories for execution by a vehicle equipped with four-wheel steering. A second vehicle model may be used to control the vehicle relative to the determined trajectories. For instance, the second vehicle model can determine leading wheels steering angles for steering leading wheels of the vehicle and trailing wheels steering angles for steering trailing wheels of the vehicle, independently of the leading wheels.

What is claimed is: 
     
       1. A vehicle comprising:
 a leading wheel disposed proximate a leading end of the vehicle; 
 a trailing wheel disposed proximate a trailing end of the vehicle; 
 one or more processors; and 
 memory storing instructions that, when executed by the one or more processors, cause the vehicle to perform actions comprising:
 receiving a reference path along which the vehicle is to travel; 
 determining, using a model of the vehicle and based at least in part on the reference path and a speed of the vehicle, a first commanded steering angle for steering the leading wheel and a second commanded steering angle for steering the trailing wheel, 
 wherein a ratio of the first commanded steering angle and the second commanded steering angle is a negative ratio when the speed is below a first threshold speed, and the ratio is a positive ratio when the speed is above a second threshold speed; and 
 controlling the vehicle based at least in part on the first commanded steering angle and the second commanded steering angle. 
 
 
     
     
       2. The vehicle of  claim 1 , the second commanded steering angle is an additive inverse of the first commanded steering angle. 
     
     
       3. The vehicle of  claim 1 , wherein the first commanded steering angle is based at least in part on a current position of the vehicle, the current position being a point of zero lateral velocity. 
     
     
       4. The vehicle of  claim 3 , wherein:
 the vehicle has a first axis associated with the leading wheel and a second axis associated with the trailing wheel; and 
 the point of zero lateral velocity is equidistant from the first axis and the second axis along a longitudinal axis extending from the leading end of the vehicle to the trailing end of the vehicle. 
 
     
     
       5. The vehicle of  claim 1 , wherein the model of the vehicle is a first model and the controlling the vehicle comprises:
 determining, based at least in part on a second model of the vehicle, a first steering angle for the leading wheel and a second steering angle for the trailing wheel, the first steering angle and the second steering angle being determined to minimize vehicle sideslip; 
 controlling, with a first steering controller, the leading wheel according to the first steering angle; and 
 controlling, with a second steering controller, the trailing wheel according to the second steering angle. 
 
     
     
       6. A method comprising:
 receiving a speed of a vehicle; 
 determining, using a model of the vehicle and based at least in part on the speed, a first commanded steering angle for a leading wheel of the vehicle and a second commanded steering angle for a trailing wheel of the vehicle, 
 wherein a ratio of the first commanded steering angle and the second commanded steering angle is a negative ratio when the speed is below a first threshold speed, and the ratio is a positive ratio when the speed is above a second threshold speed; and 
 controlling the vehicle based at least in part on the first commanded steering angle and the second commanded steering angle. 
 
     
     
       7. The method of  claim 6 , wherein the model is a kinematic model of a representative vehicle having half a wheelbase of the vehicle and constrained to have zero sideslip. 
     
     
       8. The method of  claim 6 , wherein the negative ratio decreases with decreasing speeds below the first threshold speed and the positive ratio increases with increasing speeds above the second threshold speed. 
     
     
       9. The method of  claim 6 , wherein the model is a first model and wherein controlling the vehicle comprises:
 determining, based at least in part on a second model of the vehicle, a first steering angle for the leading wheel and a second steering angle for the trailing wheel; 
 controlling, with a first steering controller, the leading wheel according to the first steering angle; and 
 controlling, with a second steering controller, the trailing wheel according to the second steering angle. 
 
     
     
       10. The method of  claim 9 , wherein the first model is a kinematic model and the second model is a dynamic model of the vehicle and determining the first steering angle and the second steering angle comprises:
 determining a first estimated steering angle for the leading wheel and a second estimated steering angle for the trailing wheel; and 
 determining, based at least in part on a difference between a state of the vehicle and a desired state of the vehicle, a first corrective steering angle for the leading wheels and a second corrective steering angle for the trailing wheels, 
 wherein the first steering angle is based at least in part on the first estimated steering angle and the first corrective steering angle and the second steering angle is based at least in part on the second estimated steering angle and the second corrective steering angle. 
 
     
     
       11. The method of  claim 10 , wherein the determining the first estimated steering angle and the second estimated steering angle is based at least in part on a steady state single track model constrained by zero sideslip for the vehicle. 
     
     
       12. The method of  claim 10 , wherein the determining the difference comprises:
 determining a lateral offset of the vehicle relative to a reference path; and 
 determining an angular offset between a heading of the vehicle and the reference path. 
 
     
     
       13. The method of  claim 12 , wherein determining the first corrective steering angle and the second corrective steering angle comprises minimizing the lateral offset and the angular offset. 
     
     
       14. The method of  claim 9 , wherein the second model is a dynamic bicycle model and the first steering angle the second steering angle are determined based at least in part on:
 one or more of a cornering stiffness of a tire comprising at least one of the leading wheel or the trailing wheel and a friction coefficient for the tire, and 
 a center of mass of the vehicle, a first distance from the center of mass of the vehicle to a first rotational axis associated with the leading wheel and a second distance from the center of mass of the vehicle to a second rotational axis associated with the trailing wheel. 
 
     
     
       15. One or more non-transitory computer-readable media storing instructions that, when executed by one or more processors, perform actions comprising:
 receiving a reference path along which a vehicle is to travel; 
 receiving a speed of the vehicle; 
 determining, using a model of the vehicle and based at least in part on the speed and the reference path, a first commanded steering angle for a leading wheel of the vehicle and a second commanded steering angle for a trailing wheel of the vehicle, 
 wherein a ratio of the first and second commanded steering angles is a negative ratio, decreasing with decreasing speeds, when the speed is below a first threshold speed, and the ratio is a positive ratio, increasing with increasing speeds, when the speed is above a second threshold speed; and 
 controlling the vehicle based at least in part on the first commanded steering angle and the second commanded steering angle. 
 
     
     
       16. The non-transitory computer-readable media of  claim 15 , wherein the model is a kinematic model and wherein controlling the vehicle:
 determining, based at least in part on a dynamic model of the vehicle, a first steering angle for the leading wheel and a second steering angle for the trailing wheel, the kinematic model and the dynamic model constraining the vehicle to have no sideslip; 
 controlling, with a first steering controller, the leading wheel according to the first steering angle; and 
 controlling, with a second steering controller, the trailing wheel according to the second steering angle. 
 
     
     
       17. The non-transitory computer-readable media of  claim 16 , wherein the determining the first steering angle and the second steering angle comprises:
 determining a first estimated steering angle for the leading wheel and a second estimated steering angle for the trailing wheel; and 
 determining, based at least in part on a difference between a state of the vehicle and a desired state, a first corrective steering angle for the leading wheel and a second corrective steering angle for the trailing wheel, 
 wherein the first steering angle is based at least in part on the first estimated steering angle and the first corrective steering angle and the second steering angle is based at least in part on the second estimated steering angle and the second corrective steering angle. 
 
     
     
       18. The vehicle of  claim 1 , wherein the negative ratio decreases with decreasing speeds below the first threshold speed and the positive ratio increases with increasing speeds above the second threshold speed. 
     
     
       19. The vehicle of  claim 5 , wherein the first model is a kinematic model and the second model is a dynamic model of the vehicle. 
     
     
       20. The non-transitory computer-readable media of  claim 17 , wherein:
 the determining the first estimated steering angle and the second estimated steering angle is based at least in part on a steady state single track model constrained by zero sideslip for the vehicle, and 
 the determining the first corrective steering angle and the second corrective steering angle comprises minimizing a lateral offset of the vehicle relative to a reference path and an angular offset between a heading of the vehicle and the reference path.